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Inverse Möbius transform of A003962.
1

%I #33 Sep 04 2023 01:59:06

%S 1,3,4,7,5,12,7,15,13,15,8,28,10,21,20,31,11,39,13,35,28,24,16,60,21,

%T 30,40,49,17,60,20,63,32,33,35,91,22,39,40,75,23,84,25,56,65,48,28,

%U 124,43,63,44,70,31,120,40,105,52,51,32,140,35,60,91,127,50,96,37,77,64

%N Inverse Möbius transform of A003962.

%F Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = (nextPrime(p)+1)/2. - _David W. Wilson_, Sep 01 2001; corrected by _Michel Marcus_, Feb 26 2015

%p f:= proc(n) local q,r,t;

%p r:= 1;

%p for t in ifactors(n)[2] do

%p q:= (nextprime(t[1])+1)/2;

%p r:= r*(q^(t[2]+1)-1)/(q-1);

%p od:

%p r

%p end proc:

%p seq(f(n),n=1..100); # _Robert Israel_, Feb 26 2015

%t f[p_, e_] := Module[{q = (NextPrime[p] + 1)/2}, (q^(e+1)-1)/(q-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Sep 04 2023 *)

%o (PARI) a(n) = {my(f=factor(n)); for (i=1, #f~, q = (nextprime(f[i, 1] + 1)+1)/2; f[i, 1] = (q^(f[i,2]+1) - 1)/(q-1); f[i, 2] = 1); factorback(f);} \\ _Michel Marcus_, Feb 26 2015

%Y Cf. A003962, A151800.

%K nonn,easy,mult

%O 1,2

%A _Marc LeBrun_

%E Corrected and extended by _David W. Wilson_, Aug 29 2001